The recently developed high-accuracy extrapolated ab initio thermochemistry method for theoretical thermochemistry, which is intimately related to other high-precision protocols such as the Weizmann-3 and focal-point approaches, is revisited. Some minor improvements in theoretical rigor are introduced which do not lead to any significant additional computational overhead, but are shown to have a negligible overall effect on the accuracy. In addition, the method is extended to completely treat electron correlation effects up to pentuple excitations. The use of an approximate treatment of quadruple and pentuple excitations is suggested; the former as a pragmatic approximation for standard cases and the latter when extremely high accuracy is required. For a test suite of molecules that have rather precisely known enthalpies of formation {as taken from the active thermochemical tables of Ruscic and co-workers [Lecture Notes in Computer Science, edited by M. Parashar (

Springer
,
Berlin
, 2002), Vol. 2536, pp. 25
38
; J. Phys. Chem. A108, 9979 (2004)]}, the largest deviations between theory and experiment are 0.52, 0.70, and 0.51kJmol1 for the latter three methods, respectively. Some perspective is provided on this level of accuracy, and sources of remaining systematic deficiencies in the approaches are discussed.

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25.

An error appears in note 62 of the original paper, in which it is stated that ROHF calculations were also used for the CN molecule. The numbers in the tables, however, are based on the ZPE calculated from UHF for CN. In this paper, the ROHF results are used.

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30.
Numerical results from all three implementations appear to agree with each other, and a sum-over-states direct summation program [
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G0=14αBeα+164kϕkkkk7576kϕkkk2ωk+364klωlϕkkl24ωk2ωl214k<l<mϕklm2ωkωlωmDklm18klαBeα(ζklα)2.
See Ref. 5 for a definition of the symbols in this equation. In Ref. 5, the fifth and sixth terms were in error. Furthermore, the kinetic energy elements in Ref. 28 were in error, which was addressed to some degree in Ref. 27.
31.

This is true, of course, for full configuration interaction wave functions and tends to be an excellent approximation for coupled-cluster methods of all types that include single excitations. Nevertheless, this assumption remains as an undesirable approximation of our original work and is rectified here.

33.
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37.

A strong Fermi resonance exists in CO2 between ν1(Σg+) and 2[ν2(Πu)], which results in a large value of G0(0.86kJmol) when calculated according to the formula in Ref. 30. However, one may alternatively skip the corresponding resonance denominator, which affects the ZPE calculated from the term level expression without G0 as well as G0 itself (the total ZPE, of course, is not affected by resonances between excited vibrational levels). In either case, the total zero-point energies are the same. The G0 contribution for CO2 in Table I is calculated with the corresponding resonance denominator omitted, as was the ZPE given in Ref. 5.

38.

Alternatives to CCSDT(Q) for the approximate treatment of quadruple excitations include the CCSDT[Q], CCSDT(Q)Λ, CCSDTQ-1a, CCSDTQ-1b, CC4, and CCSDTQ-3 schemes (see Ref. 35), of which the first two are, in fact, slightly cheaper to apply than the others. Numerical tests indicate that for total energies, the CCSDT(Q)Λ, CCSDTQ-1b, and CC4 approaches perform better than CCSDT(Q), while performance of the CCSDTQ-3 method is perhaps comparable (see Ref. 35). The CCSDT[Q] and the CCSDTQ-1a methods, on the other hand, are less accurate than any of these approaches. rms errors of the calculated heats of formation for the dataset were also evaluated using all of these methods and found to be (in kJmol1) 1.17 (CCSDT[Q]), 0.31 (CCSDT(Q)Λ), 1.12 (CCSDTQ-1a), 0.32 ( CCSDTQ-1b), 0.33 (CC4), and 0.69 (CCSDTQ-3). The conclusions concerning the performance of the approximate quadruples methods for heats of formation are similar to those for total energies; however, the difference between CCSDT(Q) and the more expensive methods is, ultimately, negligible (their use is consequently not justified). We note in passing that heats of formation were also computed using CCSDTQ as well as the aforementioned approximate methods with the cc-pVTZ basis set. However, no significant improvement has been achieved with respect to the experimental values, meaning that again the extra computational cost is not justified.

39.

It is interesting to note that both the HF-SCF and CCSD(T) energies extrapolated with the 345 sequence are more negative than their 456 counterparts, although the former extrapolation tends to give smaller atomization energies in one case (HF-SCF) and larger in the other (CCSD(T)). The latter is rather obvious and sensible: correlation energy always tends to increase atomization energies, and a method that tends to overestimate correlation energies would tend to overestimate binding energies, if the extrapolation error were somewhat systematic. However, for the HF-SCF cases, the extrapolation error (as measured by the difference between 345- and 456-based extrapolations) is larger for free atoms than those in molecules in the cases we have investigated, which is in turn responsible for the underestimated atomization energies. Differences between 345- and 456-based extrapolations(in μH) are 269 (oxygen atom), 111 (nitrogen atom), 35 (carbon atom), 22 (hydrogen atom) [atoms], 239 (OH), 109 (CN), and 190 (CO). It is interesting, indeed a bit odd, that the error for the oxygen atom is the largest.

40.
Experimental ZPE was calculated using Eq. (5) and experimental values for ωe, ωexe, the equilibrium rotational constant, Be and the rotation-vibration interaction constant, αe [
M. W.
Chase
, Jr.
,
J. Phys. Chem. Ref. Data
6
,
27
(
1998
)]..
41.
W.
Kolos
and
L.
Wolniewicz
,
J. Chem. Phys.
49
,
404
(
1968
).
42.

In many cases, vibrational frequencies are slightly overestimated in cc-pVQZ calculations, with a consequent overestimation of the zero-point energy.

43.
B.
Ruscic
(private communication); unpublished results from Active Thermochemical Tables ver. 1.25 using the Core (Argonne) Thermochemical Network ver. 1.048..
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